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Math Help - Two Dimensional Discrete Dynamical System

  1. #1
    Junior Member
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    Two Dimensional Discrete Dynamical System

    Hi, I have the following dynamical system,

    x_{n+1}={x_n}^2-{y_n}^2+a
    y_{n+1}=2x_ny_n

    Where a is real.

    And I am asked to consider the set of points on a circle of radius r and centre origin and show that they are mapped to another circle under one iteration.

    How do I go about this? I know the equation for the circle is of course x^2+y^2 = r^2. Can someone please give me a clue how to start?

    Thanks,
    Katy
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  2. #2
    Junior Member erich22's Avatar
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    Re: Two Dimensional Discrete Dynamical System

    Let
     x_0^2+y_0^2=r^2 be the initial circle to be mapped by the system.
    Calculate x_1,\text{ and } y_1, then you can find that x_1^2+y_1^2= \dots, i.e. a circle with the origin is translated along x axis.

    There you go, Katy
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